Cosmology. The Origin and Evolution of Cosmic Structure - Coles P., Lucchin F

8First Principles

It is easy to see that the Equation (1.2.8) cannot be correct in general relativity since ∂Tik/∂xk and ∂Tik/∂xk are not tensors. Since

it is evident that ∂Tmn/∂x n involves terms such as ∂2xi/∂x m∂x n, so it will not be a tensor. However, although the ordinary derivative of a tensor is not a tensor, a quantity called the covariant derivative can be shown to be one. The covariant derivative of a tensor A is defined by

in an obvious notation. The conservation law can therefore be written in a fully covariant form:

A covariant derivative is usually written as a ‘;’ in the subscript; ordinary derivatives are usually written as a ‘,’ so that Equation (1.2.8) can be written Tik,k = 0.

Einstein wished to find a relation between matter and metric and to equate Tik to a tensor obtained from gik, which contains only the first two derivatives of gik and has zero covariant derivative. Because, in the appropriate limit, Equation (1.2.12) must reduce to Poisson’s equation describing Newtonian gravity

it should be linear in the second derivative of the metric. The properties of curved spaces were well-known when Einstein was working on this theory. For example, it was known that the Riemann–Christo el tensor,

could be used to determine whether a given space is curved or flat. (Incidentally, Γkmi is not a tensor so it is by no means obvious, though it is actually true, that Rklmi is a tensor.) From the Riemann–Christo el tensor one can form the Ricci tensor:

The tensor Gik contains second derivatives of gik, so Einstein proposed as his fundamental equation

where the quantity 8πG/c4 (G is Newton’s gravitational constant) ensures that Poisson’s equation in its standard form (1.2.13) results in the limit of a weak gravitational field. He subsequently proposed the alternative form

where Λ is called the cosmological constant; as gik;k = 0, we still have Ti k;k = 0. He actually did this in order to ensure that static cosmological solutions could be obtained. We shall return to be the issue of Λ later, in Section 1.12.

1.3 The Robertson–Walker Metric

Having established the idea of the Cosmological Principle, our task is to see if we can construct models of the Universe in which this principle holds. Because general relativity is a geometrical theory, we must begin by investigating the geometrical properties of homogeneous and isotropic spaces. Let us suppose we can regard the Universe as a continuous fluid and assign to each fluid element the three spatial coordinates xα (α = 1, 2, 3). Thus, any point in space–time can be labelled by the coordinates xα, corresponding to the fluid element which is passing through the point, and a time parameter which we take to be the proper time t measured by a clock moving with the fluid element. The coordinates xα are called comoving coordinates. The geometrical properties of space–time are described by a metric; the meaning of the metric will be divulged just a little later. One can show from simple geometrical considerations only (i.e. without making use of any field equations) that the most general space–time metric describing a universe in which the Cosmological Principle is obeyed is of the form

where we have used spherical polar coordinates: r, ϑ and ϕ are the comoving coordinates (r is by convention dimensionless); t is the proper time; a(t) is a function to be determined which has the dimensions of a length and is called the cosmic scale factor or the expansion parameter; the curvature parameter K is a constant which can be scaled in such a way that it takes only the values 1, 0 or −1. The metric (1.3.1) is called the Robertson–Walker metric.

10 First Principles

The significance of the metric of a space–time, or more specifically the metric tensor gik, which we introduced briefly in Equation (1.2.2),

(as usual, repeated indices imply a summation), is such that, in Equation (1.3.2), ds2 represents the space–time interval between two points labelled by xj and xj +dxj. Equation (1.3.1) merely represents a special case of this type of relation. The metric tensor determines all the geometrical properties of the space–time described by the system of coordinates xj. It may help to think of Equation (1.3.2) as a generalisation of Pythagoras’s theorem. If ds2 > 0, then the interval is timelike and ds/c would be the time interval measured by a clock which moves freely between xj and xj + dxj. If ds2 < 0, then the interval is spacelike and |ds2|1/2 represents the length of a ruler with ends at xj and xj + dxj measured by an observer at rest with respect to the ruler. If ds2 = 0, then the interval is lightlike or null; this type of interval is important because it means that the two points xj and xj + dxj can be connected by a light ray.

If the distribution of matter is uniform, then the space is uniform and isotropic. This, in turn, means that one can define a universal time (or proper time) such that at any instant the three-dimensional spatial metric

where the interval is now just the spatial distance, is identical in all places and in all directions. Thus, the space–time metric must be of the form

This coordinate system is called the synchronous gauge and is the most commonly used way of slicing the four-dimensional space–time into three space dimensions and one time dimension.

To find the three-dimensional (spatial) metric tensor γαβ let us consider first the simpler case of an isotropic and homogeneous space of only two dimensions. Such a space can be either (i) the usual Cartesian plane (flat Euclidean space with infinite curvature radius), (ii) a spherical surface of radius R (a curved space with positive Gaussian curvature 1/R2), or (iii) the surface of a hyperboloid (a curved space with negative Gaussian curvature).

In the first case the metric, in polar coordinates ρ (0 ρ < ∞) and ϕ (0 ϕ < 2π), is of the form

we have introduced the dimensionless coordinate r = ρ/a, which lies in the range 0 r < ∞, and the arbitrary constant a, which has the dimensions of a length. On the surface of a sphere of radius R the metric in coordinates ϑ (0 ϑ π) and ϕ (0 ϕ < 2π) is just

where a = R and the dimensionless variable r = sin ϑ lies in the interval 0 r 1 (r = 0 at the poles and r = 1 at the equator). In the hyperboloidal case the metric is given by

where the dimensionless variable r = sinh ϑ lies in the range 0 r < ∞.

The Robertson–Walker metric is obtained from (1.3.4), where the spatial part is simply the three-dimensional generalisation of (1.3.5). One finds that for the three-dimensional flat, positively curved and negatively curved spaces one has,

where dΩ2 = dϑ2 + sin2 ϑ dϕ2; 0 χ π in (1.3.6 b) and 0 χ < ∞ in (1.3.6 c). The values of K = 1, 0, −1 in (1.3.1) correspond, respectively, to the hypersphere, Euclidean space and space of constant negative curvature.

The geometrical properties of Euclidean space (K = 0) are well known. On the other hand, the properties of the hypersphere (K = 1) are complex. This space is closed, i.e. it has finite volume, but has no boundaries. This property is clear by analogy with the two-dimensional case of a sphere: beginning from a coordinate origin at the pole, the surface inside a radius rc(ϑ) = aϑ has an area S(ϑ) = 2πa2(1 − cos ϑ), which increases with rc and has a maximum value Smax = 4πa2 at ϑ = π. The perimeter of this region is L(ϑ) = 2πa sin ϑ = 2πar, which is maximum at the ‘equator’ (ϑ = 12 π), where it takes the value 2πa, and is zero at the ‘antipole’ (ϑ = π): the sphere is therefore a closed surface, with finite area and no boundary. In the three-dimensional case the volume of the region contained inside a radius

12 First Principles

Figure 1.1 Examples of curved spaces in two dimensions: in a space with negative curvature (open), for example, the sum of the internal angles of a triangle is less than 180◦, while for a positively curved space (closed) it is greater.

maximum at the ‘equator’ (χ = 12 π), where it takes the value 4πa2, and is zero at the ‘antipole’ (χ = π). In such a space the value of S(χ) is more than in Euclidean space, and the sum of the internal angles of a triangle is more than π. The properties of a space of constant negative curvature (K = −1) are more similar to those of Euclidean space: the hyperbolic space is open, i.e. infinite. All the relevant formulae for this space can be obtained from those describing the hypersphere by replacing trigonometric functions by hyperbolic functions. One can show, for example, that S(χ) is less than the Euclidean case, and the sum of the internal angles of a triangle is less than π.

In cases with K ≠ 0, the parameter a, which appears in (1.3.1), is related to the curvature of space. In fact, the Gaussian curvature is given by CG = K/a2; as expected it is positive for the closed space and negative for the open space. The Gaussian curvature radius RG = CG−1/2 = a/√K is, respectively, positive or imaginary in these two cases. In cosmology one uses the term radius of curvature to describe the modulus of RG; with this convention a always represents the radius of spatial curvature. Of course, in a flat universe the parameter a does not have any geometrical significance.

As we shall see later in this chapter, the Einstein equations of general relativity relate the geometrical properties of space–time with the energy–momentum tensor describing the contents of the Universe. In particular, for a homogeneous and isotropic perfect fluid with rest-mass energy density ρc2 and pressure p, the

solutions of the Einstein equations are the Friedmann cosmological equations:

(the dot represents a derivative with respect to cosmological proper time t); the time evolution of the expansion parameter a which appears in the Robertson– Walker metric (1.3.1) can be derived from (1.3.11) if one has an equation of state relating p to ρ. From Equation (1.3.11 b) one can derive the curvature

is called the critical density. The space is closed (K = 1), flat (K = 0) or open (K = −1) according to whether the density parameter

is greater than, equal to, or less than unity.

It will sometimes be useful to change the time variable we use from proper time

1.4 The Hubble Law

The proper distance, dP, of a point P from another point P0, which we take to define the origin of a set of polar coordinates r, ϑ and ϕ, is the distance measured by a chain of rulers held by observers which connect P to P0 at time t. From the Robertson–Walker metric (1.3.1) with dt = 0 this can be seen to be

14 First Principles

Of course this proper distance is of little operational significance because one can never measure simultaneously all the distance elements separating P from P0. The proper distance at time t is related to that at the present time t0 by

where a0 is the value of a(t) at t = t0. Instead of the comoving coordinate r one could also define a radial comoving coordinate of P by the quantity

In this case the relation between comoving coordinates and proper coordinates is just

The proper distance dP of a source may change with time because of the timedependence of the expansion parameter a. In this case a source at P has a radial velocity with respect to the origin P0 given by

Equation (1.4.6) is called the Hubble law and the quantity

is called the Hubble constant or, more accurately, the Hubble parameter (because it is not constant in time). As we shall see, the value of this parameter evaluated at the present time for our Universe, H(t0) = H0, is not known to any great accuracy. It is believed, however, to have a value around

The unit ‘Mpc’ is defined later on in Section 4.1. It is conventional to take account of the uncertainty in H0 by defining the dimensionless parameter h to be H0/100 km s−1 Mpc−1 (see Section 4.2). The law (1.4.6) can, in fact, be derived directly from the Cosmological Principle if v c. Consider a triangle defined by the three spatial points O, O and P. Let the velocity of P and O with respect to O be, respectively, v(r) and v(d). The velocity of P with respect to O is

From the Cosmological Principle the functions v and v must be the same. Therefore

Redshift 15

Equation (1.4.10) implies a linear relationship between v and r:

If we impose the condition that the velocity field is irrotational,

which comes from the condition of isotropy, one can deduce that the matrix Hα β is symmetric and can therefore be diagonalised by an appropriate coordinate transformation. From isotropy, the velocity field must therefore be of the form

where H is only a function of time. Equation (1.4.13) is simply the Hubble law (1.4.6).

Another, simpler, way to derive Equation (1.4.6) is the following. The points O, O and P are assumed to be su ciently close to each other that relativistic space– time curvature e ects are negligible. If the universe evolves in a homogeneous and isotropic manner, the triangle OO P must always be similar to the original triangle. This means that the length of all the sides must be multiplied by the same factor a/a0. Consequently, the distance between any two points must also be multiplied by the same factor. We therefore have

where l0 and l are the lengths of a line segment joining two points at times t0 and t, respectively. From (1.4.14) we recover immediately the Hubble law (1.4.6).

One property of the Hubble law, which is implicit in the previous reasoning, is that we can treat any spatial position as the origin of a coordinate system. In fact, referring again to the triangle OO P, we have

which again is just the Hubble law, this time expressed about the point O .

1.5 Redshift

It is useful to introduce a new variable related to the expansion parameter a which is more directly observable. We call this variable the redshift z and we shall use it extensively from now on in describing the evolution of the Universe because many of the relevant formulae are very simple when expressed in terms of this variable.

16 First Principles

We define the redshift of a luminous source, such as a distant galaxy, by the quantity

where λ0 is the wavelength of radiation from the source observed at O (which we take to be the origin of our coordinate system) at time t0 and emitted by the source at some (earlier) time te; the source is moving with the expansion of the universe and is at a comoving coordinate r. The wavelength of radiation emitted by the source is λe. The radiation travels along a light ray (null geodesic) from the source to the observer so that ds2 = 0 and, therefore,

Light emitted from the source at te = te +δte reaches the observer at t0 = t0 +δt0. Given that f(r) does not change, because r is a comoving coordinate and both the source and the observer are moving with the cosmological expansion, we can write

If δt and, therefore, δt0 are small, Equations (1.5.2) and (1.5.3) imply that

If, in particular, δt = 1/νe and δt0 = 1/ν0 (νe and ν0 are the frequencies of the emitted and observed light, respectively), we will have

A line of reasoning similar to the previous one can be made to recover the evolution of the velocity vp(t) of a test particle with respect to a comoving observer. At time t +dt the particle has travelled a distance dl = vp(t) dt and thus finds itself moving with respect to a new reference frame which, because of the expansion of the universe, has an expansion velocity dv = (a/a)˙ dl. The velocity of the particle with respect to the new comoving observer is therefore

The results expressed by Equations (1.5.5) and (1.5.11) are a particular example of the fact that, in a universe described by the Robertson–Walker metric, the momentum q of a free particle (whether relativistic or not) evolves according to q a−1.

There is also a simply way to recover Equation (1.5.7), which does not require any knowledge of the metric. Consider two nearby points P and P , participating in the expansion of the Universe. From the Hubble law we have

where dvP is the relative velocity of P with respect to P and dl is the (infinitesimal) distance between P and P . The point P sends a light signal at time t and frequency ν which arrives at P with frequency ν at time t + dt = t + (dl/c). Since dl is infinitesimal, as is dvP , we can apply the approximate formula describing the

Doppler e ect:

The Equation (1.5.11) integrates immediately to give (1.5.5) and therefore (1.5.7).

1.6 The Deceleration Parameter

The Hubble parameter H(t) measures the expansion rate at any particular time t for any model obeying the Cosmological Principle. It does, however, vary with time in a way that depends upon the contents of the Universe. One can express this by expanding the cosmic scale factor for times t close to t0 in a power series:

is called the deceleration parameter; the su x ‘0’, as always, refers to the fact that q0 = q(t0). Note that while the Hubble parameter has the dimensions of inverse time, q is actually dimensionless.

Putting the redshift, defined by Equation (1.5.7), into Equation (1.6.1) we find that